My question, is primarily, based on this question, in which the accepted answer asserts that $\vec B \, $ has two definitions,
- As magnetic field$^*$, using Lorentz force $\vec F=q \vec v \times \vec B $
- As magnetic flux density, from $\Phi=\int\vec B.d\vec S$
My question is how do these two definitions, result in the same vector field?
$*$ I'm not able to define $\vec B$ as I could define $\vec E$, as force per unit charge. Is there one such definition for $\vec B$, or was it introduced, so that we could formulate our observations easily with such a vector field?
In laws of induction, we use B as flux density and while finding the Lorentz force, we use it as a vector field whose magnitude is given by Biot-Savart law. How are these two equal?